1. Field of the Invention
The present invention relates to a method and apparatus for the real-time measurement of ultrashort laser pulses based on frequency-resolved optical gating (FROG).
2. Related Art
Ultrashort optical pulses have very short time durations, typically less than a few 10's of picoseconds. As a result, ultrashort optical pulses are spectrally broad. Because the index of refraction of materials is a function of wavelength, different wavelengths of light travel at different speeds in optical materials causing the properties of ultrashort optical pulses to change as they propagate. The shape of the pulse can influence how the pulse itself interacts with materials, further complicating the analysis of ultrashort optical pulse transmission and propagation. Ultrafast laser pulse measurement tries to obtain both the intensity profile of the pulse as well as the “phase of the pulse,” which is the actual variation of the frequencies that make up the pulse.
Generally, time resolved measurements of events use shorter events to resolve variations in the characteristics of the events as a function of time. Unfortunately for the analysis of ultrashort optical pulses, shorter events do not exist and modern electronics are insufficiently fast to allow for direct time resolved measurements. One effective measurement technique for ultrashort optical pulses is called “frequency-resolved optical gating” or FROG. FROG is based on the observation that an optically formed spectrogram contains all of the information about the pulse to be measured. A two-dimensional phase retrieval process extracts the pulse from its spectrogram. One disadvantage of frequency-resolved optical gating is the large amount of data required to obtain all of the information needed to measure the pulse. In addition, data is needed both as a function of time and frequency (or wavelength). Aspects of the present invention improve data acquisition for FROG allowing the technique to be more versatile while retaining convenience, accuracy and speed.
Mathematical Representation of an Optical Pulse
The mathematical representation of ultrashort optical pulses is discussed to provide background for the discussion of the FROG technique. The time-dependent variations of an optical pulse are embodied in the pulse's electric field, A(t), which can be written:A(t)=Re[E(t)eiω0t]  (1)where ω0 is the carrier frequency and Re refers to the real part. A(t) can be used in this form for calculations, but it is generally easier to work with a different representation that removes the rapidly varying ω0 part, eiω0t, and uses as a representation a slowly varying envelope together with a phase term that contains only the frequency variations. This representation, which does not include the rapidly varying carrier frequency, isE(t)=[I(t)]1/2eiω(t)  (2)where I(t) and φ(t) are the time-dependent intensity and phase of the pulse. (E(t) is complex.) The frequency variation, Ω(t), is the derivative of φ(t) with respect to time:Ω(t)=−dφ(t)/dt  (3)
The pulse field can be written equally well in the frequency domain by taking the Fourier transform of equation 2:{tilde over (E)}(ω)=[Ĩ(ω)]1/2e−iφ(ω)  (4)where Ĩ(ω) is the spectrum of the pulse and φ(ω) is its phase in the frequency domain. The spectral phase contains time versus frequency information. That is, the derivative of the spectral phase with respect to frequency yields the time arrival of the frequency, or the group delay.
Obtaining the intensity and phase, I(t) and φ(t) (or Ĩ(ω) and φ(ω)) is called full characterization of the pulse. Common phase distortions include linear chirping, where the phase (either in the time domain or frequency domain) is parabolic. When the frequency increases with time, the pulse is said to have positive linear chirp; negative linear chirp is when the high frequencies lead the lower frequencies. Higher order chirps are common, but for these, differentiation between spectral and temporal chirp is required because spectral phase and temporal phase are not interchangeable.
Frequency-Resolved Optical Gating
Frequency-resolved optical gating (FROG) measures the spectrum of a particular temporal component of an optical pulse, shown in FIG. 1, by spectrally resolving the signal pulse in an autocorrelation-type experiment using an instantaneously responding nonlinear medium. As shown in FIG. 1, FROG involves splitting a pulse and then overlapping the two resulting pulses in an instantaneously responding X(3) or X(2) medium. Any medium that provides an instantaneous nonlinear interaction may be used to implement FROG. Perhaps the most intuitive is a medium and configuration that provides polarization gating.
For a typical polarization-gating configuration, induced birefringence due to the electronic Kerr effect is the nonlinear-optical interaction. The “gate” pulse causes the X(3) medium, which is placed between two crossed polarizers, to become slightly birefringent. The polarization of the “gated” probe pulse (which is cleaned up by passing through the first polarizer) is rotated slightly by the induced birefringence allowing some of the “gated” pulse to leak through the second polarizer. This is referred to as the signal. Because most of the signal emanates from the region of temporal overlap between the gate pulse and the probe pulse, the signal pulse contains the frequencies of the “gated” probe pulse within this overlap region. The signal is then spectrally resolved, and the signal intensity is measured as a function of wavelength and delay time τ. The resulting trace of intensity versus delay and frequency is a spectrogram, a time- and frequency-resolved transform that intuitively displays the time-dependent spectral information of a waveform.
The resulting spectrogram can be expressed as:
                                          S            E                    ⁡                      (                          ω              ,              τ                        )                          =                                                                        ∫                                  -                  ∞                                ∞                            ⁢                                                E                  ⁡                                      (                    t                    )                                                  ⁢                                  g                  ⁡                                      (                                          t                      -                      τ                                        )                                                  ⁢                                  ⅇ                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                                          2                                    (        9        )            where E(t) is the measured pulse's electric field, g(t−τ) is the variable-delay gate pulse, and the subscript E on SE indicates the spectrogram's dependence on E(t). The gate pulse g(t) is usually somewhat shorter in length than the pulse to be measured, but not infinitely short. This is an important point: an infinitely short gate pulse yields only the intensity I(t) and conversely a CW gate yields only the spectrum I(ω). On the other hand, a finite-length gate pulse yields the spectrum of all of the finite pulse segments with duration equal to that of the gate. While the phase information remains lacking in each of these short-time spectra, having spectra of an infinitely large set of pulse segments compensates this deficiency. The spectrogram nearly uniquely determines both the intensity I(t) and phase φ(t) of the pulse, even if the gate pulse is longer than the pulse to be measured. If the gate is too long, sensitivity to noise and other practical problems arise.
In FROG, when using optically induced birefringence as the nonlinear effect, the signal pulse is given by:Esig(t,τ)∝E(t)|E(t−τ)|2  (10)So the measured signal intensity IFROG(ω,τ), after the spectrometer is:
                                          I                          F              ⁢                                                          ⁢              R              ⁢                                                          ⁢              O              ⁢                                                          ⁢              G                                ⁡                      (                          ω              ,              τ                        )                          =                                                                        ∫                                  -                  ∞                                ∞                            ⁢                              E                ⁡                                  (                  t                  )                                                                          ⁢                      E            ⁡                          (                              t                -                τ                            )                                ⁢                                                                    2                            ⁢                                                ⅇ                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                                  ⁢                                  ⅆ                  t                                                                    2                                              (        11        )            The FROG trace is thus a spectrogram of the pulse E(t) although the gate, |E(t−τ)|2, is a function of the pulse itself.
FROG is not limited to the optical Kerr effect. Second harmonic generation (SHG) FROG can be constructed to analyze relatively weak pulses from oscillators and is typically more sensitive than polarization-gating FROG. For SHG FROG, the pulse is combined with a replica of itself in an SHG crystal as illustrated in FIG. 2.
To see that the FROG trace essentially uniquely determines E(t) for an arbitrary pulse, note that E(t) is easily obtained from Esig(t, τ). Equation (11) can then be written in terms of Esig(t, Ω), the Fourier transform of the signal field Esig(t, τ) with respect to delay variable τ. This gives the following, apparently more complex, expression:
                                          I                          F              ⁢                                                          ⁢              R              ⁢                                                          ⁢              O              ⁢                                                          ⁢              G                                ⁡                      (                          ω              ,              τ                        )                          =                                                                        ∫                                  -                  ∞                                ∞                            ⁢                                                                    E                    sig                                    ⁡                                      (                                          t                      ,                      Ω                                        )                                                  ⁢                                  ⅇ                                                                                                              -                          ⅈ                                                ⁢                                                                                                  ⁢                        ω                        ⁢                                                                                                  ⁢                        t                                            -                                              ⅈ                        ⁢                                                                                                  ⁢                        Ω                        ⁢                                                                                                  ⁢                        τ                                                              )                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                ⁢                                  ⅆ                  Ω                                                                          2                                    (        12        )            Equation (12) indicates that the problem of inverting the FROG trace IFROG(ω, τ) to find the desired quantity Esig(t, τ) is that of inverting the squared magnitude of the two-dimensional (2-D) Fourier transform of Esig(t, τ). This problem, which is called the 2-D phase-retrieval problem, is well known in many fields, especially in astronomy, where the squared magnitude of the Fourier transform of a 2-D image is often measured. At first glance, this problem appears unsolvable; after all, much information is lost when the magnitude is taken. The 1-D phase retrieval problem is known to be unsolvable (for example, infinitely many pulse fields give rise to the same spectrum). Intuition fails badly in this case, however. Two- and higher-dimension phase retrieval processes essentially always yield unique results.
FROG Inversion
An iterative 2-D phase retrieval process is used to extract the pulse information from the measured FROG trace as illustrated generally in FIG. 3. This phase retrieval process converges to a pulse that minimizes the difference between the measured and the calculated FROG trace. Application of this phase retrieval process to FROG has been problematic in the past. Some recent applications use a generalized projections algorithm, which converges quickly, along with faster computers, to track pulse changes at rates of 20 Hz or greater, making FROG a real-time pulse measurement technique. Indeed, programs for analyzing FROG traces are commercially available.
The original FROG inversion process, using what is commonly referred to as the vanilla algorithm, is simple and iterates quickly. On the other hand, the process tends to stagnate and give erroneous results, especially for geometries that use a complex gate function such as SHG or self-diffraction. Improved strategies using different algorithms, including brute force minimization, were developed to avoid stagnation, at the expense of both iteration speed and convergence time. Later a numerical technique called generalized projections brought a significant advance in both speed and stability. The generalized projections technique proceeds after an iteration by constructing a projection that minimizes the error (distance) between the FROG electric field, Esig(t, τ), obtained immediately after the application of the intensity constraint, and the FROG electric field calculated from the mathematical form constraint. The projection constructed in this manner is used as the starting point of the next iteration.
The first implementations of the generalized projections technique used a standard minimization procedure to find the electric field for the next iteration (which can still be slow). For the most common FROG geometries, PG and SHG, there are substantial advantages to a different strategy that directly determines the starting point for the next iteration. This strategy, called Principal Components Generalized Projections (PCGP), converts the generalized projections technique into an eigenvector problem. The PCGP technique has achieved pulse characterization rates of 20 Hz.
The goal of phase retrieval is to find the E(t) that satisfies two constraints. The first constraint is the FROG trace itself which is the magnitude squared of the 1D Fourier transform of Esig(t,τ):
                                          I                          F              ⁢                                                          ⁢              R              ⁢                                                          ⁢              O              ⁢                                                          ⁢              G                                ⁡                      (                          ω              ,              τ                        )                          =                                                                        ∫                                  -                  ∞                                ∞                            ⁢                                                                    E                    sig                                    ⁡                                      (                                          t                      ,                      τ                                        )                                                  ⁢                                  ⅇ                                                            -                      ⅈ                                        ⁢                                                                                  ⁢                    ω                    ⁢                                                                                  ⁢                    t                                                  ⁢                                                                  ⁢                                  ⅆ                  t                                                                          2                                    (        13        )            The other constraint is the mathematical form of the signal field, Esig(t,τ), for the nonlinear interaction used. The mathematical forms for a variety of FROG beam geometries are:
                                          E            sig                    ⁡                      (                          t              ,              τ                        )                          ∝                  {                                                                                          E                    ⁡                                          (                      t                      )                                                        ⁢                                                                                                          E                        ⁡                                                  (                                                      t                            -                            τ                                                    )                                                                                                            2                                                                                                P                  ⁢                                                                          ⁢                  G                  ⁢                                                                          ⁢                  F                  ⁢                                                                          ⁢                  R                  ⁢                                                                          ⁢                  O                  ⁢                                                                          ⁢                  G                                                                                                                                                E                      ⁡                                              (                        t                        )                                                              2                                    ⁢                                                            E                      *                                        ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    S                  ⁢                                                                          ⁢                  D                  ⁢                                                                          ⁢                  F                  ⁢                                                                          ⁢                  R                  ⁢                                                                          ⁢                  O                  ⁢                                                                          ⁢                  G                                                                                                                          E                    ⁡                                          (                      t                      )                                                        ⁢                                      E                    ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    S                  ⁢                                                                          ⁢                  H                  ⁢                                                                          ⁢                  G                  ⁢                                                                          ⁢                  F                  ⁢                                                                          ⁢                  R                  ⁢                                                                          ⁢                  O                  ⁢                                                                          ⁢                  G                                                                                                                                                E                      ⁡                                              (                        t                        )                                                              2                                    ⁢                                      E                    ⁡                                          (                                              t                        -                        τ                                            )                                                                                                                    T                  ⁢                                                                          ⁢                  H                  ⁢                                                                          ⁢                  G                  ⁢                                                                          ⁢                  F                  ⁢                                                                          ⁢                  R                  ⁢                                                                          ⁢                  O                  ⁢                                                                          ⁢                  G                                                              }                                    (        14        )            where PG is polarization gate, SD is self-diffraction, SHG is second harmonic generation and THG is third harmonic generation FROG.
All FROG implementations work by iterating between two different data sets: the set of all signal fields that satisfy the data constraint, IFROG(ω, τ), and the set of all signal fields that satisfy equation 14. The difference between FROG implementations is how the iteration between the two sets is completed. In the case of generalized projections, the E(t)'s are chosen such that the distance between the E(t) on the magnitude set and the E(t) on the mathematical form set is minimized. This is accomplished by minimizing the following equation:
                    Z        =                              ∑                          i              ,                              j                =                1                                      N                    ⁢                                                                                                        E                                          sig                      ⁡                                              (                                                  D                          ⁢                                                                                                          ⁢                          C                                                )                                                                                    (                      k                      )                                                        ⁡                                      (                                                                  t                        i                                            ,                                              τ                        j                                                              )                                                  -                                                      E                                          sig                      ⁡                                              (                        MF                        )                                                                                    (                                              k                        +                        1                                            )                                                        ⁡                                      (                                                                  t                        i                                            ,                                              τ                        j                                                              )                                                                                      2                                              (        15        )            where Esig(DC)(k)(ti,τj) is the signal field generated by the data constraint, and Esig(MF)(k+1)(ti,τj) is the signal field produced from one of the beam geometry equations 14. For the normal generalized projections technique, the minimization proceeds using a standard steepest decent algorithm; the derivative of Z with respect to the signal field is computed to determine the direction of the minimum. The computation of the derivatives is tedious; the derivatives are tabulated in, for example, Trebino, et al., Rev. Sci Instrum., 68, p. 3277 (1997).
An alternative to the minimization of equation 15 is principal components generalized projections (PCGP). PCGP computes the starting point of the next iteration through an eigenvector problem, reducing the computation for the next iteration to simple matrix-vector multiplies. PCGP works for both the PG and SHG beam geometries, is simple to program and is fast, as described in D. J. Kane, IEEE J. Quant. Elec., (1999).
Self Checks in FROG Measurements
Unlike other pulse measurement techniques, FROG can provide a great deal of feedback about both the quality of the measurement (systematic errors) and the quality of the technique's performance. A good check for convergence is the FROG trace error together with a visual comparison between the retrieved FROG trace and the measured FROG trace. The FROG trace error is given by:
                    G        =                                            1                              N                2                                      ⁢                                          ∑                                  i                  ,                                      j                    =                    1                                                  N                            ⁢                                                                                                                                    I                                                  F                          ⁢                                                                                                          ⁢                          R                          ⁢                                                                                                          ⁢                          O                          ⁢                                                                                                          ⁢                          G                                                                    ⁡                                              (                                                                              ω                            i                                                    ,                                                      τ                            j                                                                          )                                                              -                                          α                      ⁢                                                                                          ⁢                                                                        I                                                      F                            ⁢                                                                                                                  ⁢                            R                            ⁢                                                                                                                  ⁢                            O                            ⁢                                                                                                                  ⁢                            G                                                                                (                            k                            )                                                                          ⁡                                                  (                                                                                    ω                              i                                                        ,                                                          τ                              j                                                                                )                                                                                                                                      2                                                                        (        16        )            where α is a renormalization constant, IFROG is the measured FROG trace and IFROG(k)(ω, τ) is computed from the retrieved electric field. Typically, the FROG trace error of a PG measurement should be less than 2% for a 64×64 pixel trace, while the FROG trace error of a 64×64 pixel SHG FROG trace should be about 1% or less. Acceptable FROG trace errors decrease as FROG trace size increases and increase for smaller FROG traces. These values are only guidelines and variations are expected in specific circumstances. For example, acceptable phase retrievals of large and very complicated FROG traces can produce larger FROG trace errors.
In a good FROG measurement, the spectrum of the retrieved pulse should faithfully reproduce the salient features of the pulse's measured spectrum. SHG FROG even provides an additional check called the frequency marginal. The sum of an SHG FROG trace along the time axis yields the autoconvolution of the pulse's spectrum providing two ways the FROG measurement can be checked. First, the autoconvolution of an independently measured spectrum can be compared to the sum of the FROG trace along the time axis (the frequency marginal) providing an indication of how well the measurement was made. For example, if the doubling crystal was too thick in the pulse measurement, the FROG trace's frequency marginal will be narrower than the autoconvolution of the measured spectrum. Second, comparing the autoconvolution of the retrieval pulse spectrum with the FROG trace marginal can provide a test of convergence in addition to a test of the measurement. Phase matching problems show up as a mismatch between the FROG trace marginal and the autoconvolution of the retrieved spectrum.
Because FROG is a spectrally resolved autocorrelation, summing any FROG trace along the frequency axis yields the autocorrelation of the measured pulse. This autocorrelation can be compared to an independently measured autocorrelation, or a comparison can be made between the frequency sum of the FROG trace and the autocorrelation calculated from the retrieved pulse to determine algorithm convergence and the quality of the measurement.
Making FROG Real-Time
In order to make FROG a real-time pulse measurement technique, the FROG device, the data acquisition, the 2-D phase retrieval process, and the user interface are integrated. One real-time ultrashort laser pulse measurement implementation combined a multishot FROG device with two digital signal processing (DSP) boards in a fast personal computer. This implementation is described in D. J. Kane, IEEE J. Quant. Elec., (1999). One DSP card was devoted to data acquisition and the other DSP card was used to make the calculations. Briefly, as an optical delay line was scanned, the signal from the nonlinear interaction was spectrally resolved. Thus, temporal slices of the FROG trace, or spectrogram were obtained. Appropriate (sequential) ordering and assembly of the temporal slices formed the FROG trace.
The DSP implementation was considered expensive for some applications and so efforts were directed to develop a FROG-based ultrashort laser pulse measurement that did not use DSP cards. One implementation that did not use a DSP, called VideoFROG, used a standard CCD video camera to acquire the FROG trace and a standard computer frame grabber to digitize the video signal. The computer displayed the resulting FROG trace and resized and conditioned the FROG trace for the phase retrieval process. The computer also was the user interface, displaying the retrieved pulse and the raw video image. VideoFROG used the PCGP phase retrieval process, which generally converges well. Consequently, no provision was made to detect stagnation or other faults in the phase retrieval process.
Basic Controllers—Proportional Integral Derivative Control
The simplest and fastest controller is just the basic proportional controller. However, these most basic controllers are prone to calibration errors and offsets. The measured difference will not correspond to the actual difference and the pulse will never achieve the desired pulse shape—the system will have a steady-state error in the set point (Dorf and Bishop, Modern Control Systems, p. 193)
A simple calculation illustrates this effect. The schematic shown in FIG. 4 depicts a typical control loop. Suppose R(s) is the transfer function (Laplace transform) for the input to the pulse shaper, Y(s) is the transfer function for the output, E(s) is the transfer function for the error, Gc(s) is the controller transfer function, G(s) is the transfer function of the servo itself, and H(s) is the transfer function for the position encoder or sensor. The output of the system is given by (Dorf and Bishop, Modern Control Systems, p. 178-180):
                              Y          ⁡                      (            s            )                          =                                                                              G                  c                                ⁡                                  (                  s                  )                                            ⁢                              G                ⁡                                  (                  s                  )                                                                    1              +                                                                    G                    c                                    ⁡                                      (                    s                    )                                                  ⁢                                  G                  ⁡                                      (                    s                    )                                                  ⁢                                  H                  ⁡                                      (                    s                    )                                                                                ⁢                      R            ⁡                          (              s              )                                                          (        17        )            The error of the system, which is the difference between the measured pulse and the desired pulse (R(s)), is given by (Dorf and Bishop, Modern Control Systems, p. 183):
                              E          ⁡                      (            s            )                          =                              1                          1              +                                                                    G                    c                                    ⁡                                      (                    s                    )                                                  ⁢                                  G                  ⁡                                      (                    s                    )                                                  ⁢                                  H                  ⁡                                      (                    s                    )                                                                                ⁢                      R            ⁡                          (              s              )                                                          (        18        )            For purposes of an example, in the case of proportional control only, suppose we set Gc(s)=K, H(s)=1 and G(s)=1 (ideal case). Thus,
                              E          ⁡                      (            s            )                          =                                            1                              1                +                K                                      ⁢                          R              ⁡                              (                s                )                                              =                                    1                              1                +                K                                      ⁢                          1              s                                                          (        19        )            when the input is a unit step function (R(s)=1/s). To calculate the steady state error, we use the final-value theorem (Dorf and Bishop, Modern Control Systems, p. 183) which is:
                                          lim                          t              ->              ∞                                ⁢                      (                          ⅇ              ⁡                              (                t                )                                      )                          =                              lim                          s              ->              0                                ⁢                      (                          sE              ⁡                              (                s                )                                      )                                              (        20        )            where E(s) is the Laplace transform of e(t). Thus, we find that the steady-state value of the error, e(t) is 1/(1+K).
Therefore, a simple proportional feedback control system is unacceptable for the level of precision and accuracy required by ultrafast laser pulse measurement.
One way to address all of the control issues is to include terms in addition to the proportional feedback term. For example, an integral term will remove steady-state errors. An integral controller has a transfer function of (Dorf and Bishop, Modern Control Systems, p. 703):
                                          G            c                    ⁡                      (            s            )                          =                              K            p                    +                                    K              I                        s                                              (        21        )            where KI is the integral gain term. Setting G(s) and H(s) to 1 and solving for E(s), we get:
                              E          ⁡                      (            s            )                          =                              s                                                            K                  p                                ⁢                s                            +                              K                I                                              ⁢                      1            s                                              (        22        )            Applying the final-value theorem as mentioned above, we find that as t→∞, the offset (error), e(t)→0. Unfortunately, the addition of an integral term does come at a cost—a reduction of the transient response and an increase in the overshoot. A derivative term is usually added to improve transient response, decrease overshoot, and decrease the settling time. Thus, the simplest form of a robust, high precision, high accuracy controller with good speed is the proportional-integral-derivative, or PID, controller (FIG. 5):